Theorem iscmgmALT 47053

Description: The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ismgmALT.b	⊢ 𝐵 = (Base‘𝑀)
ismgmALT.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
iscmgmALT	⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵))

Proof of Theorem iscmgmALT

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		fveq2 6881	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3	1, 2	breq12d 5151	. . 3 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ (+g‘𝑀) comLaw (Base‘𝑀)))
4		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	breq12i 5147	. . 3 ⊢ ( ⚬ comLaw 𝐵 ↔ (+g‘𝑀) comLaw (Base‘𝑀))
7	3, 6	bitr4di 289	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ ⚬ comLaw 𝐵))
8		df-cmgm2 47049	. 2 ⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
9	7, 8	elrab2 3678	1 ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5138  ‘cfv 6533  Basecbs 17142  +_gcplusg 17195   comLaw ccomlaw 47014  MgmALTcmgm2 47044  CMgmALTccmgm2 47045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-cmgm2 47049

This theorem is referenced by: (None)